ELSEVIER

Electrical Power & Energy Systems, Vol. 18, No. 5, pp. 279-295, 1996 Copyright © 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0142-0615(95)00066-6 0142-0615/96/$15.00 + 0.00

Hard-limit induced chaos in a fundamental power system model

W Ji a n d V V e n k a t a s u b r a m a n i a n School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99164-2752, USA wji@eecs.wsu.edu, mani@eecs.wsu.edu

The paper investigates complex nonlinear phenomena in a fundamental power system model represented in a single- machine infinite-bus formulation. The generator electro- magnetics, electromechanics and its excitation control are modelled together by fourth-order differential equations. It is shown that when excitation control gains are set high (as in common industry practice) and when the excitation hard-limits are taken into account, this representative power system model undergoes global bifurcations includ- ing period-doubling cascades which lead to sustained cha- otic behaviour. Specifically sustained complex oscillations result from the interaction of hard-limits and the system transients over a large range of realistic parameter values. The emergence of strange attractors is demonstrated in the paper by detailed numerical simulations and preliminary analysis. Copyright © 1996 Elsevier Science Ltd.

Keywords: power system dynamics, power system stability, bifurcations, chaos, hard limits

I. I n t r o d u c t i o n For studying the interaction of a single generator (or a generation unit more generally) with a large power system, it is customary to model the inter-connected power system in an equivalent single-machine infinite- bus (called SMIB in short) representation I. While such models are of limited validity for studying instability events that involve a large number of generators, the dynamic phenomena observed in this simple SMIB power system model do provide significant insight into what can be expected from the interaction of one generator with the rest of the power system. Moreover, such SMIB systems do occur in the real power system under special circumstances. The advantage of the SMIB representation is the simplicity of the model which facilitates sophisti- cated nonlinear analysis. However, caution must be exer- cised in interpreting the results for the physical system.

Received 13 April 1995; accepted subject to revisions 29 June 1995; accepted 17 August 1995

This paper studies the dynamics of an SMIB model which makes minimal but representative provisions for the angle dynamics (generator electromechanics), the voltage dynamics (electromagnetics) and a voltage control device (excitation field control). The resulting fourth-order dynamical system is then a reasonable and quite realistic dynamic model for studying nonlinear phenomena emerging from the interaction of the excit- ation voltage control and the generator dynamics. The simplicity of the model allows us to carry out a detailed analysis of the local and global bifurcation phenomena. Specifically, the paper proves the emergence of sustained chaotic behaviour involving a strange attractor over realistic parameter values.

Recently there has been great co,,:ern in the power industry on possible occurrences of voltage related instabil- ity events 2-4. To tackle these voltage problems, an emerg- ing control practice in the industry is to set the control gains of the excitation voltage controls at very high gain values, aimed at tighter voltage profiles and faster voltage responses. However, such high excitation control gains introduce problems of their own such as an increased participation of excitation hard-limits in voltage instabil- ity phenomena. Many recent industrial reports point to the need for a better understanding of the role of hard- limits (especially the excitation hard-limits) in voltage

4 events . Specific instability phenomena involving hard- limits have been studied in References 5 and 6 and a detailed exposition to their modelling concepts can be seen in Reference 7. A recent paper s makes a preliminary effort towards developing a formal stability theory for analysing the various types of hard-limits including limits on actuation signals and states as well as limits that involve relay switchings. Several new concepts and analyti- cal results are presented in Reference 8 which are primarily motivated by hard-limit related phenomena in power system models.

The role of a wind-up type hard-limit 7'8 present in the excitation control is studied in this paper. We show that global dynamic phenomena are drastically affected by the presence of the excitation hard-limits. For instance, after an instability event when the local stability of the

279

280 Hard-limit induced chaos in a fundamental power system modeL W. Ji and V. Venkatasubramanian

operating point gets destroyed at a subcritical Hopf bifurcation 9 (essentially a Hopf bifurcation occurs when a simple pair of eigenvalues crosses the imaginary axis upon certain parameter variation such as the power- generation in Section III), the resulting transient diverges away leading to a system breakdown when the hard- limits are not modelled in the analysis. However, when the hard-limits (which are always present in the physical system) are taken into consideration, the hard-limits contribute some 'positive damping' in a global sense so that the transients converge to a stable limit cycle leading to sustained oscillations. Moreover, this stable limit-cycle undergoes further global bifurcations such as period doubling cascades that eventually lead to the birth of a strange attractor and chaos.

This paper for the first time proves the occurrence of sustained chaotic behaviour induced by excitation hard- limits in a representative SMIB power system model over realistic parameter values.

While the emergence of oscillations from hard-limit interactions has been long-studied in the control engi- neering context (for instance the method of describing functions is motivated by these concerns), the occurrence of such hard-limit induced oscillations from the inter- actions of excitation controls and subcritical Hopf bifur- cations was first observed in Reference 10. A recent paper 11 studies a related problem of hard-limit induced oscillations analytically in a two-dimensional system sub- ject to state limits.

Chaotic behaviour in power system models has been investigated by several papers. References 12 and 13 analysed the chaotic motions in swing equations by using the Melnikov's technique. Reference 14 uses perturbation techniques to analyse resonance and chaos in a single- machine quasi-infinite bus power system. References 15 and 16 analysed global bifurcation phenomena in an algebraically constrained power system model and showed the significance of complex bifurcations in voltage stabil- ity analysis. Reference 17 postulated a dynamic mecha- nism for voltage collapse from tlie dynamics on the centre manifold of a saddle-node bifurcation and this paper introduced a simple power s~stem model which has been extensively investigated by several recent papers 18-21. References 18, 19 and 20 studied the power system model in Reference 17 and showed that the chaotic behaviour results through a period-doubling bifurcation route. Reference 21 investigated the global bifurcations and chaos in the same model under different parameters and showed the presence of chaos through period doub- ling bifurcations. Reference 21 also established a new dynamic mechanism for the voltage collapse from the evolution of a 'boundary crisis' when the strange attractor collides with a saddle-point leading to the disappearance of the strange attractor at a blue-sky bifurcation.

This paper focuses on the dynamics of a simple SMIB power system model. Local bifurcations in the model are analysed and global bifurcations such as the period- doubling bifurcations are studied numerically through phase-portraits, time-plots and Lyapunov exponents. Explicit numerical simulations show the emergence of chaotic behaviour through the period-doubling cascade route and the eventual destruction of the strange attrac- tor. Complex phenomena involving global bifurcations such as period doubling bifurcations of the limit cycles are introduced rather intuitively through explicit numeri- cal simulations. The occurrence of the chaotic behaviour

persists over large parameter variations hence these phe- nomena can be readily simulated. Moreover, since cha- otic behaviour of this paper is observed in a realistic model over practical parameter values, we believe that such chaotic phenomena can possibly be experienced in the real power system.

The organization of this paper is as follows. Section II introduces the SMIB model briefly and sets up the state space and the parameter space. Section III analyses local bifurcation phenomena by computing the bifurcation curves in the parameter space. Section IV explores global phenomena in the absence of the hard-limits in the field excitation control, and Section V considers nonlinear phenomena and chaotic behaviour when the hard-limit is included in the analysis.

II. An S M I B power system model Consider the simple power system shown in Figure 1 which represents the classical single-machine infinite-bus (SMIB) power system 1 . This representation is commonly used in power system dynamic studies for identifying and understanding instability problems which principally involve one generation unit.

The generator electromagnetics or the angle dynamics can be represented by the following swing equations

= ~0co (1)

M ~ = - D w + (PT -- PG) (2)

where M = 2H is the moment of inertia in seconds; COo = 27rf0 -- 27r60 = 377 rad/s; D is the damping factor in per unit; PT represents the mechanical power input in p.u.; and PG is the electrical power generated in p.u.; 6 is the machine angle (rad) and w is the machine frequency deviation in p.u.

By assuming that the damper windings are not present, the generator electromagnetics can be modelled by the single axis flux decay equation

' " = - E ' T d o E + (Xd -- Xd)I d 4- Efd (3)

where xa is the synchronous reactance, x~ is the transient reactance, T~0 is the direct axis transient time-constant. Here E t denotes the internal flux decay or the voltage magnitude behind the transient reactance, Efd is the field excitation voltage, and Id is the direct axis armature current. For the single-machine infinite-bus system shown in Figure 1, we can easily derive

E t - 1 • cos 6 I d - (4)

x + X'd

so that we can rewrite (3) as f

X d - - X d , -, Xd + X E' -+ cos ~ 4- Efd (5) T do E -- I ! Xd .-t- X Xd ..t- X

Next we assume that the excitation field control is present in the generator and is simply represented by

E'/8 V/0 jx 1L0

Figure 1. The single-machine infinite-bus power model

Hard-limit induced chaos in a fundamental power system modek W. Ji and I/. Venkatasubramanian 281

V

Vref

K A

I+STA

Figure 2. The simplified excitation model

Efd0

+ ~--;Efdrl E'dm~--/ E edr

[ Efdmi n E fdmax

the single-time-constant transfer function shown in Figure 2. Note that in general, generator models and the models for the excitation controls will be far more detailed and the models stated so far in this paper amount to 'minimal' yet representative models for turbogenerator units 1 . In Figure 2, Vre f is the reference bus voltage; Efd 0 is the reference field voltage or the DC set-point of the field control; and the voltage', V refers to the bus voltage at the generator bus terminal .and for the SMIB representation, it is easily computed as

1 V/[(Xtd + xE' cos 6) 2 + (xE' s in 6) 2] (6) V - x + x----7 d

Specifically note the presence of the wind-up type hard- limit on the signal Efdr which limits the control output Efd to be strictly between the values Efdm,. and Efdm,x.

With all these assumptions, the model of the SMIB system for stability analysis can be stated together as

E: 6 = 27rf0w (7)

E I Ma) = -Dw + PT x, d + ~ sin 6 (8)

I

T~o/~, _ x d + x E' + xd -- xd COS 6 -[- Efd (9) x +x

TA/~fd r = - K A ( V - Vref) - (Efd r -- Efd0) (10)

where V has been defined above in equation (6) and Efd is the output of the wind-up limiter

e Efam.x if Efdr>Efdm. x

Efd = Efd r if gfdml n ~ gfd r ~ Efdma x (11)

Ef0m," if Efdr < Efd~,

The state space for this power system now consists of four state variables 6, w, E ' and Efd. Let us denote the state variables together as the state vector X where X = (6, w, E', Efdr) T. Among the parameters, the syn- chronous machine parameters M, Xd, x~ and T~0 do not vary during system operation (ignoring saturation effects), and the transmission line parameter x can also be treated as a constant for our purposes. Among the remaining parameters, the mechanical power input PT and the voltage set-point Vref are the primary operating parameters for controlling the active power (frequency control) and reactive power (voltage control) respec- tively. The excitation control time-constant TA is usually treated as a constant while the control gain KA may be varied for special control actions. The damping para- meter D models the overall damping effects in this simple representation and can be modelled as an operating parameter which summarizes the damping effects of other elements that contribute towards damping in the actual system.

In this paper, we primarily investigate the system dynamics E for variations in the power parameter PT

under high control gain values for K A. However, for illustrating some phenomena numerically, it is more convenient to treat the damping constant D as a bifur- cation parameter and hence D is also used in some of the bifurcation diagrams as an additional parameter variation. This essentially amounts to taking a different cross-section of the parameter space (by varying D instead of P-r) for a nicer illustration of the phenomena. We want to emphasize that identical phenomena (such as period doubling bifurcations and the birth of strange attractors) can be observed under load PT variations by considering suitable parameter values as shown later in the paper. For instance, the evolution of the strange attractor is demonstrated in Section V for variation in PT with D = 2.

In summary, the states 6, w, E' and Efdr constitute the state space, and the parameters PT and D are the bifur- cation parameters for our analysis. The other parameters in the SMIB system E are chosen to be

H = 5 M = 10 T~o = 10 T A = 1

X d = 1 x~ = 0.4 x = 0.5 Vre f = 1.05

E f d 0 = 2 Efdm~" = 0 g f d m . x = 5 K A = 190

which are typical values for these parameters. Note that the excitation control gain KA has been chosen as KA = 190 which corresponds to a moderately high gain value for the system shown in Figure 2. The global bifurcation phenomena observed in this paper can indeed be verified numerically over a diverse set of other parameter values. However, for simplifying the presentation, we restrict the analysis in this paper to the set of values shown above.

II1. Analysis of local bifurcations In this section, we analyse the effects of parameter vari- ations in PT and D on the system behaviour locally near equilibria, specifically the structure of local bifurcations in E, Section III.1 establishes the static equilibrium structure under PT variation while Section III.2 deals with local bifurcation phenomena.

II1.1 Equilibrium points We want to study the equilibrium structure of ~ under variations in PT and D. While solving for the equilibrium points of the system, note that the damping factor D does not affect the static structure of the equilibrium points. Hence our problem reduces to considering only the effects of the power PT on the equilibrium computations. For small PT, it is clear from engineering intuition that the field control Efdr value at the operating point must lie within the linear region of the hard-limit, i.e. between gfdm~, and Efd=~. In fact, the steady state value of the control output Efd r should always lie within these limiting values in a well-designed control system even under load

282 Hard-limit induced chaos in a fundamental power system model. W. Ji and V. Venkatasubramanian

variations and we make this as a preliminary assumption to start the analysis. For our chosen set of parameter values, it can indeed be verified that the hard-limits of the wind-up limiter (11) are never reached at the operating point. Therefore we can substitute Efd ---= Efd r in equation (9) in equilibrium computations. Thus the equilibrium point for N is the solution of the following set of equations

0 = 2~rfow (12)

E' - - sin6 (13) O = - D w + P.r XPd + X

O - - Xd -t'- X E t nt Xd -- Xtd C O S 6 + E f d (14) X'd + X X'd + X

0 = - K A ( V - Vref) - (Efd -- Efd0) (15)

For our chosen parameters, the above equation can be rewritten as

a~ = 0 (16)

Efd = 5 E ' - 2 cos6 (17)

0.9PT = E ' sin 6 (18)

V = 1 . 0 5 ~ E ' - Z c ° s 6 - 2 KA (19)

where

V = ~ ; [ 1 6 + 25(E') z + 40E' cos6]

Now since KA has been chosen to be a high gain value (KA = 190), the second part on the right side of equation (19) can be omitted to approximate the terminal voltage as V ~ Vref = 1.05. Note that this is a standard approxi- mation in steady state analysis of the power system and essentially amounts to the concept of the PV-bus for those buses with generation units 1. In our case, the approximation can be explicitly justified by the inequality

5 E ' - ~cos 6 - 2 K A = 5 ( E ' - 1.2)_3_K_A-- 2cos6

IE' - 1.21 + 0.4 < << 1.05 - 104

Therefore equations (12)-(15) can be simplified as

= o ( 20 )

Efd = S E ' - 2 cos6 (21)

E ' sin 6 = 0.9PT (22)

E ' cos 6 -- 73.3025 - 25(E ' ) 2 (23) 40

After further simplification, we get

(E') 4 - 8.4242(E') 2 + 8.59721 + 2.0736p2T = 0 (24)

It is easily seen that this quadratic equation in E t2 has positive real solutions whenever Pa" lies within the region -2.1 < PT < 2.1. Therefore the parameter value PT ---- 2.1 delineates the static boundary or the static saddle-node bifurcation. Ignoring the negative values, if PT < 2.1, the above equation (24) will have two positive solutions for E ' and thus the system ~ will have two equilibrium

points. If Pv>2.1, the above equation (24) will not have any solution so that the original system E does not have any equilibrium point. Hence PT = 2.1 app, oximates a saddle-node bifurcation point for our system E.

To reflect on the accuracy of the approximation that V ~ Vref = 1.05, i.e. on the validity of (24) for equi- librium solution, we compute the two equilibrium points Xh and X1 of the system E numerically at PT = 1.3 from (12)-(15) which are

Xh = (1.0409, 0, 1.3559, 1.9229) T

X~ = (2.6621,0, 2.5362, 4.8184) T

Indeed these values compare well with the approximate results obtained from (24) as

X h = (1.0416, 0, 1.3554, 1.9224) a"

XI = (2.6682, 0, 2.5665, 4.87079) 1"

111.2 Local bifurcations

When the parameters PT and D are varied, the operating point undergoes changes in its location and the corre- sponding system eigenvalues of the linearized Jacobian matrix (evaluated at the operating point) will move around in general. From a local stability view-point, it is important to understand those parameter values when the system eigenvalues cross the imaginary axis since these parameter values then outline the boundary of feasible operation at the operating point 22. Specifically the parameter values where the local stability at the operating point undergoes changes are local bifurcations and these bifurcations pinpoint the parameter values where the limit-set or the attractor (that corresponds to the current operating condition) could change from a stable equilibrium point to other limit sets such as stable periodic orbits.

For 2, as we only have four differential equations in the model with no algebraic equation constraints, the case of the unbounded eigenvalues or the singularity induced bifurcation of Reference 22 is ruled out and there exist only two possible imaginary axis crossings. These are the imaginary axis crossings either (1) at the origin, or (2) on the imaginary axis away from the origin. The former case (1) of zero eigenvalues generically leads to static bifurca- tions or the saddle node bifurcations, and the latter case (2) of purely imaginary eigenvalues corresponds to Hopf bifurcations generically 9.

We first consider the saddle-node bifurcation where the determinant of the linearized Jacobian matrix must be zero to admit zero eigenvalues. Since the damping param- eter D does not affect the equilibrium points (is not present in the determinant equation either), it does not have any effect on the saddle-node bifurcation. There- fore, the saddle-node bifurcation is a vertical wall in the D direction in the parameter space. Hence, we only need to consider one parameter Pr in this case. Numerically we can compute the saddle-node bifurcation point with respect to Pa" which turns out to be Pa'sn = 2.0826. Note that the saddle node value of PT -- 2.0826 agrees very well with the approximate computation of the saddle node (in Section III.1) as Pa- ~ 2.1.

Next let us consider the Hopf bifurcation points where the system Jacobian matrix has a pair of purely imaginary eigenvalues. In this case, both parameters D and Pa" do affect the bifurcation computation and the bifurcation

Hard-limit induced chaos in a fundamental power system model." W. J i and V. Venkatasubramanian 283

350

3 0 0

2 5 0

2 0 0

1 5 0

100

50

I

Hopf bif u ~ i i t i i I

S a d d l e - - n o d ~

B i f u r c a t i o n

i 1.2 1.4 ' 1 16 1 1.8 2 P T

2.2

Figure 3. The bifurcation point of the system in the parameter space

equations turn out to be quite messy, preventing any explicit analysis. However, the Hopf bifurcation locus can be computed numerically and the results are shown in Figure 3 for variations in Pa" and D together with the saddle-node bifurcation locus (which is a vertical line in the D direction as explained above).

We observe from Figure 3 that the Hopf bifurcation locus for E divides the parameter space into two regions: (1) the region above (and to the left of) the Hopf bifurcation curve in FiLgure 3 is the feasiblity region 22 or the operating region of the system where the operating point (the equilibrium point Xh) is locally stable, and (2) the region below (and to the right of) the Hopf bifurca- tion locus in Figure 3 where the equilibrium point Xh is locally unstable. Moreover, for the parameter values in the region to the right of the saddle node bifurcation locus in Figure 3, no equilibrium point exists.

We need to make a remark on the range of the damping parameter D shown in Figure 3. Note that D values as shown range from D := 0 to D = 350. As D is stated as per unit values in our representation, in practice, the parameter D typically can range from D = -10p.u. to D = 20 p.u. The damping constant could be somewhat higher, say up to D = 50 in highly damped systems, and the values above say D = 100 may be mainly of academic interest. However, in our analysis and in Figure 3, we display a larger window of D parameters. This larger window of D values here is aimed at establishing the location of the more degenerate bifurcations known as

9 higher codimension bifurcations such as those from the intersection of the saddle-node and Hopf bifurcations. These degenerate bifurcations can have a drastic effect on global bifurcation phenomena and hence on global dynamic properties even when they occur far away from the actual locations of these bifurcations. Essentially the degenerate local bifurcations can act as the 'sources' of interesting global limit-sets and are hence important in any bifurcation study 9'23'24. Owing to space limitations, we will not discuss these issues in more detail in this paper.

Moving on to the analysis of Hopf bifurcations, we claim that the Hopf bifurcations encountered in later sections are always subcritical (see Section V). Therefore for the fixed D parameter value, as the power loading parameter PT is increased, unstable limit cycles surround- ing the operating point shrink onto the equilibrium at the Hopf bifurcation point (on the Hopf bifurcation locus in

Figure 3) so that the equilibrium becomes locally unstable for higher PT values. As such, these subcritical Hopf bifurcations then provide no indication on the existence of stable limit cycles and stable attractors near the locally unstable operating point after the occurrence of the subcritical Hopf bifurcation. In the next two sections we observe that the emergence of stable limit cycles is related to the presence of the wind-up limiter, in the excitation control of Erd so that these limit cycles are born out of strictly global mechanisms.

IV. Divergence in the absence of hard- limits For the sake of completeness, and for emphasizing the significance of excitation hard-limits in determining global dynamic phenomena, we first present a set of trajectory simulations by ignoring the presence of excit- ation hard-limits. In other words, in this section, we assume that E f d m i n = --0(3 and E f d m ~ * = -q-~. In practice, an excitation control has a rich set of hard-limits for protecting the control from excessive control actions, and ignoring the control hard-limits may not be practically justifiable. And interestingly enough we will see in this section that such hard-limits actually enforce a 'regu- larity' on global dynamic phenomena. That is, there do not appear to be any stable attractors outside the feasi- bility region (region below and to the right of the Hopf bifurcation locus in Figure 3) for the system where the hard-limits are not modelled. However, in Section V, when the hard-limits are incorporated into the analysis, the transients diverging away from the unstable focus approach stable attractors which exist over large para- metric domains outside the feasibility region. This includes the existence of strange attractors for certain parameter values. Therefore, hard-limits contribute significantly to global dynamic phenomena and we second the recom- mendations in recent modelling studies 4 that hard-limits need to be treated in more detail in future analytical studies.

For the sake of brevity, we present only two numerical simulations of system trajectories in Figures 4 and 5, when the hard-limits are not modelled. Figure 4 is for the case when P'r = 1 and D = 2 and Figure 5 shows the case when PT = 1.3 and D = 110. The two points correspond to two different points outside the feasibility region where the equilibrium point is locally unstable (with a pair of

284 Hard-limit induced chaos in a fundamental power system mode/. W. Ji and V. Venkatasubramanian

¢0

P T = I , D = 2

5

0

- - 5

1 5 0 . 1

0 . 0 5

N o

--0.05 0 5 1 0 1 5 0 5 1 0 1 5

2 0

0

- - 2 0

- - 4 0

2 . 5

ELI

1 . 5

1 0 5 1 0 1 5 0 5 1 0 1 5

Figure 4. Divergence in the absence of the hard-limit

°iooi / 5° I

O 0 5 ' 0 1 O 0 1 ' 5 0 t

4

3

0

--1 0

P T = I , 3 , D = 1 1 0

o o;t 0.02

- - 0 . 0 2 0 0 0

1°i f w

- - 5 0

- - 1 O 0 5 0 1 O 0 1 5 0 2 0 0

t

Figure 5. Divergence in the absence of the hard-limit

eigenvalues in the open right half-plane). In both cases, the trajectory appears to be simply diverging away to infinity so that system protection would interfere to break up the system eventually.

Interestingly in this case, the system is locally unstable at the equilibrium point Xh for the power value PT = 1.3 which is very much lower than the static limit PT = 2.08 (see the saddle-node bifurcation at PT = 2.08 in Figure 3) even with an extremely high damping value such as D = 110. This case then points to the need for studying the imaginary axis crossings and the Hopf bifurcations in computing maximum loadability limits for PT-

When the hard-limits are introduced in the next sec- tion, we will see that for these parameter values PT -- 1.3 and D = 110, there exist stable operating conditions which are, however, sustained oscillations. And for higher PT values, the stable limit cycles undergo global bifurca- tions on to the emergence of chaotic behaviour, and interestingly all these global dynamic phenomena occur for power values PT well below the static power limit PT = 2.08.

V. Global phenomena wi th hard-l imits In this section, we incorporate in the analysis the wind-up hard-limits for our system ~ as stated in equations (11). When the wind-up hard-limits on Efd are introduced, it turns out that the diverging trajectories (diverging away

- j 5 0 1 0 0

t 1 5 0 2 0 0

from the locally unstable equilibrium) after the subcriti- cal Hopf bifurcation will eventually converge to a stable limit cycle with a rather large region of attraction for the limit cycle. In other words, the hard-limits somehow introduce positive damping in a global ,sense so that the trajectories are eventually stabilized into sustained oscil- lations. We will study these phenomena in more detail in the remainder of this section.

Let us choose Pa" = 1.3 and vary the damping param- eter D for a preliminary investigation of the dynamic phenomena. For the power value PT = 1.3, the system operating point is given by

Xo = (60, Wo, E~, E f d o ) T = (1.0409, 0, 1.3559, 1.9229) T

(25) The linearized Jacobian matrix of the system is stable at this operating point if the damping factor D > D H =

120.073. At D = D n = 120.073, the eigenvalues of the Jacobian matrix are

A] = -8.0619, )~2 = -5.1121, A3, 4 = -4-j3.3901

so that the system undergoes a Hopf bifurcation at D = D n. By using the centre manifold theorem 25, the Hopf normal coefficient a can be computed as a = 0.0500 (the details are omitted because of space limitations). As a > 0, this Hopf bifurcation at D = DH is in fact a sub- critical Hopf bifurcation 9 so that an unstable limit cycle for D > DR shrinks on towards the equilibrium as D is

0 5 0 1 0 0 1 5 0 2 0 0 t

Hard-limit induced chaos in a fundamental power system modeL W. J i and V. Venkatasubramanian 285

P T = I , D = 2

1

0 . 6

X 1 0 - 3 4

,i 4

0 5 0 1 0 0 0 5 0 1 0 0 t t

1 . 2 8 I , l | l J ~ 1 . 2 6

l=.Fr,. umalllnllmll|HilnimHiilll I 3

1

0

--1

~lhllllihiiijiltllii~llHilJJh~liijiillil/li~iillihJlJllii~lliJIt

0 5 0 1 0 0 0 5 0 1 0 0 t t

Figure 6. Sustained oscillations with the hard-limit

P T = 1 . 3 , D = 1 1 0 x 1 0 - 3

1

1.1

1

0.9

0.8 0

0 . 5

¢" o

- - 0 . 5

- -1 5 0 1 0 0 1 5 0 2 0 0 0 5 0 1 0 0 1 5 0 2 0 0

t t

1 . 4 5

1 . 4

" = 1 . 3 ~

1 . 2 ~

1 . 2 0

- - , 5 0 1 0 0 1 5 0 2 0 0 0 5 0 1 0 0 1 5 0 2 0 0

t t

Figure 7. Sustained oscillations with the hard-limit

decreased and the equilibrium loses its local stability at D = DIj. For D <DH there does not exist any stable attractor locally near the unstable equilibrium for the case of subcritical Hopf bifurcation 9. The details on the Hopf bifurcation phenomena will not be discussed here as they are well-understood in the literature (e.g. References 26, 27, 15 and 28). Instead, we will carefully look at the dynamic phenomena after the Hopf bifurcation (for D < DH, PT = 1.3) to understand the global dynamics of E.

V. 1 Persistence of hard-limit induced sustained oscillations First we will show that unlike the case in Section IV (when the hard-limits were ignored), there exists a stable limit cycle as an attractor for the system with hard-limits. Time-plots of the trajectories for the system with the hard-limit is presented in Figures 6 and 7 for the same initial condition as in Figures 4 and 5 and with respec- tively the same set of parameters (PT = 1, D = 2) and (PT = 1.3, D = 110). From the Era time-plots in Figures 6 and 7, we observe that the field voltage Efd gets clipped at the lower limit g f d m i n = 0 in both cases, so that the hard- limit does participate in generating the sustained oscil- lations. Local stability of the limit cycle can be proved formally by computing the Floquet multipliers 9 or by computing the Lyapunov exponents and we will present details on the Lyapunov exponents later in this paper (see Section V.5).

Note that from a practical view-point, sustained oscil- lations induced by hard-limits demand a careful scrutiny, in the sense that the participation of hard-limits over long time-periods during a transient event could eventually lead to protective relay action (for instance, the thermal over-heating relays might interfere). However, we will not discuss these issues in this paper.

Let us continue our observations based on numerical simulations at the moment and we decrease D from D = 110 to more realistic values such as to the value D = 20 in Figure 8. In Figure 8, we note that as the damping D is decreased, the oscillations grow in magni- tude. However, the limit cycle remains stable and sus- tained oscillations emerge as possible operating conditions of the system. Figure 8 shows the limit cycles in the 6 - ~ cross-section for several different D values, all with the same initial condition (1.0399, -0.0010, 1.3549, 1.9219) 7.

These simulations show that the limit cycle grows in size when damping D is decreased, but the stable limit cycle persists even when the parameters are varied over a large region and also when the initial conditions are over a large region in the state space. Hence we claim that hard-limit induced sustained oscillations persist over a range of parameters with seemingly large regions of attractions in the state space. Therefore 'robust' stable oscillations persist as suitable operating conditions for D < D H even after the subcritical Hopf bifurcation at D = DH in this SMIB power system.

286 Hard-limit induced chaos in a fundamental power system model. W. J i and V. Venkatasubramanian

X 1 0 - 3 D = 8 0

X 10 -3

° if 1.2

d e l t a D = 4 0

0 . 8 1 1.2 d e l t a

x 10 ̀ 3 D = 6 0

--6 O18 1 112 d e l t a

x 1 0 ` .3 D = 2 0

~ ° N _

6 0 . 8 1 1 .2

d e l t a

Figure 8. Persistence of the hard-limit induced sustained oscillations

V.2 Period-doubling cascade--the route of chaos In this section, we will see that the stable periodic orbits observed in Section V. 1 undergo a sequence of period- doubling bifurcations (the 'period doubling cascade') as the damping parameter D is decreased, which eventually leads to the birth of chaos. The emphasis in this section is on illustration through numerical simulation plots, and a proof for the existence of chaos is presented by the computation of Lyapunov exponents in Section V.5. All the results in this section are identified and are demon- strated through simple numerical simulation of trajec- tories (using Matlab, to be precise). Hence we believe that the analysis presented in this section based on ~-co phase portraits of the limit cycles can be readily carried over to far more detailed power system models by using standard transient computation programs. Results in this direction which show the evolution of chaos in more detailed power system models will be presented elsewhere.

The existence of period-doubling bifurcations in the SMIB model is justified by the sequence of limit cycles shown in Figure 9 (as 6-ca cross-sections) for various D

values. Intuitively we claim from the pictures that there exists a period-doubling bifurcation between any two successive lower D values in Figure 9 where the frequency of the periodic orbit gets halved (or equivalently where its period gets doubled) (see the respective time-plots of the angle variable ~ in Figure 10).

For instance, note the 'double twist' of the limit cycle in going from D = 9.6 to D = 2.6 suggesting the occurrence of a period-doubling bifurcation (or a subharmonic bifur- cation or a flip bifurcation 9) between D = 9.6 and D = 2.6. Furthermore, the trajectory appears to go around four times before emerging as a periodic orbit for D-- 1.7, indicating that the frequency has been further halved or that there is another period doubling bifurcation between D = 2.6 and D = 1.7. In fact, as we decrease D further, the successive occurrences of period-doub•g bifurcations get closer and closer (9.6 to 2.6 to 1.7 to 1.59 .. . etc.) and with lower values of D, the stable periodic orbit gets more and more complicated with 'more twists' or 'more wind- ings'. The period of the limit cycle is doubled each time by the occurrence of a period-doubling bifurcation so that

G.) E o

x 10 -3 D = 9 . 6 4

2

0

- 2

- 4 0 .5 1

d e l t a

D = 1 . 5 9

¢o

E o

.5

0.01

0 . 0 0 5

0

- 0 . 0 0 5

- 0 . 0 1 0

D = 2 . 6 D = 1 . 7 0 .02

0.01

o3

0 E o

- 0 . 0 1

- 0 . 0 2 1 2 0 1 2

d e l t a d e l t a

D = 1 . 5 5 D = 1 . 5 2

0.01

0 . 0 0 5 o~

0 E o

- 0 . 0 0 . ~

- 0 . 0 "

0.01

0 . 0 0 5 oB cr}

0

- 0 . 0 0 5

- 0 . 0 1

(> - 0 . 0 1 5 - 0 . 0 1 5

0 1" 2 0 1 2 d e l t a d e l t a

0.01

0 . 0 0 5

o 0 E o

- 0 . 0 0 5

--0.01

- 0 . 0 1 5 0 1 2

d e l t a

Figure 9. Period-doubling cascade in the power system

Hard-limit induced chaos in a fundamental power system model." W. J i and V. Venkatasubramanian 287

D = 9 . 6

0 10 20 30 40 t

D=1.7

0 20 40 60 80 t

D=1.55 , , , ,

0 20 40 60 80 t

D=2.6 2

0 0 10 20 30 40

t D=1.59

0 20 40 60 80 t

D=1.52

2 ' "

0 50 100 150 t

Figure 10. Some time domain plots of period-doubling cascade

the period of the limit cycle increases geometrically; for some D value near D - - 1.48, the period doubling cascade reaches its accumulation point (period T tends to infinity) that leads to the birth of a strange attractor and chaos as a climatic event. In the system E, chaos appears to exist over a range of parameter values for D before it is destroyed by other global events as we see later in Section V.3.

To understand better the mechanism of a period- doubling bifurcation, we will first indulge in a brief technical discussion. The stability of a periodic orbit can be characterized by defining a return map or a Poincar~ map for a certain cross-section across some point on the periodic orbit and by computing the eigen- values or the Floquet multipliers of this Poincar6 map 9. In other words, the Floquet multipliers can be thought of as the eigenvalues for an associated discrete system that is related to a point on the periodic orbit. When all the Floquet multipliers of a periodic orbit lie within the unit circle, the trajectories in the vicinity of the periodic orbit are strictly moving towards the periodic orbit and the periodic orbit is locally attractive. Under parametric variations, if any of the Floquet multipliers moves out of the unit circle, then the periodic orbit undergoes a change in its stability status and such parameter values where the periodic orbit has Floquet multipliers on the unit circle correspond to roughly the periodic orbit equivalents of the local bifurcations of equilibrium points. However, periodic orbits being one (or higher) dimensional objects, the bifurcations where the Floquet multipliers lie on the unit circle are much richer in terms of global dynamic phenomena compared with local bifurcations of equilibrium points.

Of special interest here is the bifurcation when one

Floquet multiplier crosses the unit circle at - 1 at a bifurcation point which then leads to a flip bifurcation or a period-doubling bifurcation. Essentially we can prove (under certain transversality conditions and by assuming the supercritical case 9) that the stability of the original periodic orbit undergoing the flip bifurcation is destroyed while there emerges a new periodic orbit which is stable and is of twice the period as that of the original periodic orbit.

Returning to our simulations in Figure 9, the phenom- enon just described precisely leads to the doubling of the periodic orbit at some D value between D = 9.6 and D = 2.6. Note that such a period-doubling bifurcation then also results in another unstable periodic orbit (the destabilized version of the original orbit encountered at D = 9.6) which is not shown here and is more difficult to locate numerically (as it is not an attractor). Therefore, in each step in Figure 9, as we go towards lower D values, we observe further period-doubling bifurcations and we have to keep in mind that at each step, an unstable periodic orbit is being generated together with a stable periodic orbit of twice the period. Moreover, each of these orbits may undergo further bifurcations. Therefore, as we near tile accumulation point of this period-doubling cascade, intuitively we see a complex region where possibly there exist a very large number of stable periodic orbits as well as unstable periodic orbits all crowded together in a closed domain resulting in 'contractions' along certain directions in this domain while 'stretching' occurs in some other directions within the domain. Moreover the cascad- ing effect of these bifurcations implies that these orbits could possibly become a continuum in some sense which can now be visualized as a complex limit set or the strange attractor.

288 Hard-limit induced chaos in a fundamental power system model. W. J i and V. Venkatasubramanian

V.3 Chaos As the period-doubling bifurcations go on, the orbits of the system get more and more complicated so that near the accumulation point of the period-doubling cas- cade, the fate of the periodic orbit is unclear. Figure 11 shows the time-plots of the voltage E t for an initial condition X1 where XI is near the unstable focus X0 at

X1 = X0 - (0.001,0.001,0.001,0.001) T (26)

For D = 1.5. Note that there appears an intermediate 'burst' in Figure 11 (resembling an 'intermittency' mechanism 9) which renders the behaviour of the system quite complex for D = 1.5.

Next, for D = 1.49, Figure 12 shows the phase portrait of a single trajectory for D = 1.49 and there appears to be

a complex attracting set (much like the 'double scroll attractor' of the Lorenz system9). We observe from this phase plot that the system trajectory while 'straying' freely within a specific region in the phase space is, however, bounded within this region and does not diverge away. Clearly the motion within this region is quite complex.

With further reduction in damping, as a result of the period-doubling cascade, the system runs into a chaotic region where the system behaves irregularly. Figures 13 and 14 show the simulation results for the case D = 1.4845 and D = 1.48425 for identical initial condi- tions X1, so that the irregularity of the system behaviour can be seen clearly from these simulation results. More- over note that the phenomenon persists over very large

1

u J

1

O 1 O O 2 0 0 3 0 0 t

Figure 11. Voltage-time plot for D = 1.5

1 . 7

1 . 6

1.5,

1 .4 ,

1 .3 ,

2 . 5

D = 1 . 4 g

delta 0 \ ' 0 0 . 0 1 5 0 . 0 1 0 . 0 0 5

omega

Figure 12. Complexity of a trajectory for D = 1.49

- 0 . 0 0 5 - 0 , 0 1

\

- 0 . 0 1 5

w

Figure 13. Voltage-time plot for D = 1.4845

Hard-limit induced chaos in a fundamental power system model: W. J i and V. Venkatasubramanian 289

2 .5

D = 1 . 4 8 4 2 5

Figure 14. Voltage-time plot for D = 1.48425

time-intervals (shown here up to 600 s) suggesting the presence of some attracting set.

Next we will analyse the properties of this attracting set as related to chaotic motions. To prove the existence of a chaotic motion 9, we will compute the Lyapunov expo- nents of the attractor in Section V.5 and show that there exists a positive Lyapunov exponent which is generally taken as a measure of chaotic systems. In the next two subsections, we first illustrate the well-known character- istic of chaotic motions, namely the sensitive dependence on initial conditions and then show the existence of a complex attractor possibly a strange attractor.

V.3.1 Sensitive dependence on initial conditions Let us choose D = 1.485 and let X0 denote the normal operating point of the power system introduced before. We simulate the system with the two initial conditions: (1) X1 defined in (26); and (2) X" 2 which is a very small perturbation of Xt

X2 ---- X 1 -F (10 -7, 10 -7, 10 -7, I0-7) T (27)

The time simulation results are shown in Figure 15 for the

angle variable 6. We observe here that while the system responses appears to resemble each other for X1 and )(2 at first, soon chaotic dynamics leads to independent, widely divergent trajectories. There is no resemblance between these two responses say for t>150s. Essentially, in Figure 15, the initial phase (roughly t < 150 s) corre- sponds to the convergence to the strange attractor while the later phase ( t> 150 s) is the motion on the strange attractor where even slightly different initial conditions such as X1 and X2 can result in two very different trajectories.

As a second illustration, we repeat the simulations for D = 1.48425, again with the initial conditions: (1))(1; and (2) X2, and a comparison of the results are shown in Figure 16. The sensitivity with respect to the initial con- dition in this case is even more clear than in Figure 15. Almost the same initial condition results in absolutely different system behaviours so that the behaviour of the system cannot be predicted without infinitely many bits of accuracy on the initial condition zg. Moreover the 'initial phase' when the trajectories apparently converged to the strange attractor (observed over the time-period t < 150 s

0 0

Ini t ial c o n d i t i o n x l

2

1 .5

0 .5

I

5 0

2 .5

2

1 .5

-o 1

0 .5

0 0

m I I I I ' I

100 150 2 0 0 2 5 0 3 0 0 3 5 0 t

Ini t ial c o n d i t i o n x 2 i

4il IulglUIIIIMIdd L /ml

I I

1 O0 150

i

ill i i I

2 0 0 t

Figure 15. Sensitive dependence on initial conditions for D = 1.485

i i

I

2 5 0 3 0 0 3 5 0

290 Hard-limit induced chaos in a fundamental power system model: W. Ji and V. Venkatasubramanian

2.5 Init ial cond i t ion x l

2 4 0.5

0

1.5 ¢1) -o 1

2.5

M I

100

N I I

150 200 t

i i I

250 300

Init ial cond i t ion x2

a~ 1.5 Q )

"o 1 1/ 0.5 iii i i I

1 O0 I

150

ill I

200

m I

250 300

Figure 16. Sensitive dependence on initial conditions for D = 1.48425

D=1.48425, x l D=1.48425, x2

1.8

1 4 ]

1.20 , ~ , ~ , , . ~ 0 . 0 1 0 " 0 2

o

omega delta 3 -0 .02

.... ~.~ T, '~.

1.8 ¢ ..... ~"

~2~,. /o .o2 o -..,~ Z,/-o.ol

" ~ o m e g a delta 3 --0.02

Figure 17. Two trajectories moving on the strange attractor

in Figure 15 for D = 1.485) is much shorter in Figure 16 (over the time-period t < 80 s approximately) for the new parameter D = 1.48425.

As the initial conditions are the same in Figures 15 and 16, the shorter duration of the initial convergence phase implies that the strange attractor is growing in size as the parameter is varied from D = 1.485 to D = 1.48425 in Figures 15 and 16 respectively.

V.3.2 Strange attractor We have shown that the system response in the sense of time-simulations or trajectories is quite sensitive to the initial condition within the chaotic region so that it seems impossible to predict the exact behaviour of the system in the time domain. However, in this section, we show that there still appears to be some 'regularity' among the irregularity, pointing to the existence of an attracting set or a stable attractor. This stable set becomes more evident when we look at the phase portraits (instead of

time-plots) as shown in Figure 17 of the trajectories starting from the two initial conditions X1 and X2 for D = 1.48425.

Although we observed in Section V.3.1 that the time- plots of these two trajectories are quite different (see Figure 16), the two trajectories appear to follow quite similar patterns in the phase space of Figure 17. This precisely indicates the existence of a strange attractor, i.e. the trajectories of the system are quickly attracted to a complex set wherein every orbit is unstable in some sense 29. The chaotic behaviour of the system is caused by these two counter-actions within the system, i.e. the system is stabilized away from the attracting region so that an orbit is quickly absorbed into the region and is destabilized within the region where no regular 'ordinary' limit set exists.

V.3.3 Divergence and system break-up When the damping parameter D is decreased below

Hard-limit induced chaos in a fundamental power system modeL" W. Ji and V. Venkatasubramanian 291

D = 1.48425, the strange attractor grows larger and larger so that it may eventually 'hit' or collide with another unstable limit set such as a saddle point or an unstable limit cycle, resulting in the annihilation of the strange attractor at a boundary crisis 2~ .

The exact mechanism of the disappearance of the attractor or the crisis is not clear to us at this point. Reference 21 analysed a boundary crisis from the col- lision of the strange attractor with a saddle point and the crisis was shown to lead to system voltage collapse.

The crisis in the SMIB model E also evolves when the strange attractor seemingly hits an unstable limit set resulting in the disappearance of the attractor at a blue- sky bifurcation. The aftermath of the boundary crisis can be seen from the phase portrait shown in Figure 19 for D = 1.484 where the strange attractor has disappeared as a stable attractor. Note that this trajectory displays transient chaotic behaviour (which is considered the 'signature' of a crisis event) by initially staying for a long time within the 'remnants' of what used to constitute the strange attractor in the state space before eventually diverging away. Moreover, the divergence is monotonous and appears to be along the unstable manifold of the saddle point X1 indicating the influence of the saddle X1 in the crisis event. However, the attractor as a whole appears to stay away from the saddle point X1 which implies the involvement of a different limit set possibly an unstable limit cycle in the crisis event. Note that for this parameter value D = 1[.484, all initial conditions would lead to diverging trajectories and system collapse. How- ever, even slightly different initial conditions would lead to very different time-domain outcomes in the sense that the durations of the preliminary transient chaotic phases (before the eventual divergence) can be quite different.

The exact nature of the dynamics along the diverging transient would determine the fate of the system after the crisis such as whether the transient reaches some other stable limit set, or the transient diverges away resulting in system break-up. The simulations in Figure 18 indicate that for the SMIB model, the machine angle 6 increases monotonously along the trajectories for parameter values after the boundary crisis so that the generator under study would lose synchronism possibly leading to system break-up. From Figure 19, we observe that the internal voltage E' in the SMIB model is better behaved along the divergence while the machine angle quickly diverges away. Therefore, the instability phenomenon in this case appears to be strictly an angle instability problem.

V.4 Persistence of the phenomena Throughout Section V.3, the power parameter PT was kept at PT = 1.3 while the damping parameter D was varied to illustrate the existence of the chaotic dynamics. A natural question that arises is whether the observations in Section V.3 are simply isolated phenomena or whether they occur persistently. For instance, we can pose the question whether these phenomena can be seen by vary- ing the loading parameter PT which may be of more interest from the practical point of view. In this section, we take a horizontal cross-section of the parameter space in Figure 3 by varying PT while keeping D = 2 at a constant value.

Even though chaotic attractor exists over very small parameter PT intervals in this section, We point out that the existence of chaotic behaviour has been shown in this paper along an arbitrary D cross-section (by varying D while PT = 1.3) in Section V.3, and along an arbitrary PT cross-section (by varying PT while keeping D = 2) in this

D = 1 . 4 8 4 , P T = I . 3

2

1 . 6

1

0 . 5

0 0

: , i J ~ '

5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0

Figure 18. Quick divergence of machine angle

D = I . 4 8 4 , P T = 1 . 3

,t . 8

1 . 7

1 . 4

1 . 3

1 . 2 o 1 2 3 4 5 6 7 d e l t a

Figure 19. Divergence and loss of synchronization

292 Hard-limit induced chaos in a fundamental power system model," W. J i and V. Venkatasubramanian

~ 0.02 t

oE m 001 !

0

PT=I PT=1.2

0

~ 0'02 t

°i E 0

- 0 . 0 ' 0.5 1 1.5 2 0 0.5 1 1.5 delta delta

PT=1.31 PT=1.32

2

-0020 02 1 1.2 2 -0020 o12 1 2

~ 0,02 t

m 00i !

o

delta delta PT=1.323 PT=1.32325

~ 0.02 t

m 001 !

- - . 0.5 1 1.5 2 0 0.5 1 1,5 2 delta delta

Figure 20. Period-doubling cascade

section. Therefore Sections V.3 and V.4 together justify our claim that chaotic behaviour exists in our SMIB model over a rather wide range of practical parameter values. Moreover, the precursor to the chaotic phenom- ena namely the cascade of period-doubling bifurcations develops over a range of parameter values. These period- doubling bifurcations introduce more frequency content at each step (frequency of the oscillations gets halved at each bifurcation so that the oscillations become more and more complex) so that the implications of the chaotic phenomena can indeed be seen (and 'felt') over large parametric domains.

Let us choose the damping D to be D = 2 and study the power PT variation. When the power PT is increased, it is easily seen that the equilibrium point denoted Xh remains locally stable up to PT = 0.9725. The system undergoes a subcritical Hopf bifurcation at PT = 0.9725 (the Hopf normal form coefficient a can be computed as a = 0.3709 > 0) and for PT > 0.9725, the equilibrium point is no longer small-signal stable, even though as observed in Section V.3, there do exist hard-limit induced stable limit cycles which qualify as possible operating conditions for the model ~. In Figure 20, we present a set of 6-~ phase- plots which show the evolution of these stable limit cycles through a sequence of period doubling bifurcations exactly like the case of D variation in Section V.3.

We can see clearly from Figure 20 that the dynamic phenomena are identical in the case of D variation: (1) for PT = 1, a stable limit cycle exists induced by the hard- limits (the Hopf bifurcation is again subcritical); (2) at PT = 1.2 the stable periodic orbits has doubled its period indicating a period doubling bifurcation between PT = 1 and PT = 1.2; (3) further period doubling bifurcations occur between every two successive PT values from PT = 1.2 to PT = 1.31 to PT = 1.32 . . . etc.; (4) clearly the proximity of the successive values in this sequence indicates that the period doubling cascade happens which

results in the birth of chaos. Indeed chaotic behaviour generated in this system by the period-doubling cascade can be proved by the computation of Lyapunov expo- nents in Section V.5. Here, we present two time-plots of the machine angle 6 for two power values PT = 1.32325 and PT = 1.32328 to demonstrate the irregular nature of the time-domain response in Figure 21.

When PT is further increased, the attractor collides with another unstable limit set leading to a boundary crisis. For higher load values, say when PT = 1.3233, the trajectories while exhibiting transient chaotic behaviour eventually diverge away implying the annihilation of the strange attractor at the boundary crisis.

It must be pointed out that all these dynamic events involving sustained oscillations and sustained chaotic beha- viour occur for power values PT near PT = 1.32 which is well below the static power limit or the saddle node bifurcation at PT = 2.08. Therefore sustained oscillations exist as candi- dates for system operating points over a large parametric domain when PT lies in the interval 0.9725 < PT < 1.3233 even though the oscillations become more complicated as the power increases owing to the occurrence of the period- doubling bifurcations eventually becoming 'irregular' or chaotic for PT near PT ---- 1.32. A stable attractor does not exist as a feasible operating condition for power values PT > 1.3233, and hence the system collapse is inevitable in the model E for PT > 1.3233 and D = 2.

Sustained chaotic oscillations associated with the strange attractor contain a wide range of frequencies and such wide frequency spectrum could cause perma- nent equipment damage (if present over a long period of time). Moreover, the irregular nature of chaos could pose serious problems in frequency control studies. Therefore the existence of chaotic behaviour in this representative power system model over realistic parameter values empha- sizes the need for understanding such phenomena more clearly in detailed power system models.

Hard-limit induced chaos in a fundamental power system model." W. Ji and V. Venkatasubramanian 293

2 . 5

2

(o 1 .5

-o 1

0 . 5

0

2 . 5

P T = 1 . 3 2 3 2 5 i i

0 1 O0

1 .5 ¢P

-o 1 201

0.5

O0 5 0

I I I

1 5 0 2 0 0 2 5 0 3 0 0 t

P T = 1 . 3 2 3 2 8

I I I

1 0 0 1 5 0 2 0 0 t

Figure 21. Chaotic behaviour of the system

I

2 5 0 3 0 0

As a final remark, the, authors claim that all the results on global phenomena so far in this paper in fact can be verified for other cross..sections of D and PT parameter values, and also for diverse values of other system param- eters. Essentially the complex nonlinear phenomena studied in this paper appear to correspond to fundamental prop- erties of the SMIB system E introduced in Section II. However, the task of analytically proving these properties promises to be quite challenging.

V.5 Computation of the Lyapunov exponents As we can see from the simulation of this paper, the motion of the trajectories of the system becomes more and more complicated as the period-doubling cascade goes along, so that the chaotic motion after the accumu- lation point of the period-doubling cascade is so complex that it is difficult to follow the evolution of the chaotic motion simply by numerical simulations. Hence, we compute the Lyapunov characteristic exponents in this section to confirm the simulation observation already made in this paper.

The Lyapunov characteristic exponents are a general- ization of the concept of eigenvalues which summarize the information on ~Lhe average contractions and separations along a certain trajectory over a long time-period 9. Strictly speaking, Lyapunov exponents (their signs basically) can be used to determine the stability status of any limit-set or any limiting beha- viour. However, practically Lyapunov exponents are often used to assess the stability properties of an attrac- tor or an attracting limit set.

Let us briefly revise the definition of Lyapunov exponents 29. Consider the nonlinear differential equation

= f ( x ) , X(to) = Xo (28)

Substitute the solution (bt(xo, to) for x into (28) so that

~t(Xo, to) = f(c~t(Xo, to), t), Ot(Xo, to) = Xo (29)

Next differentiate with respect to the initial condition x0 and denote ~t(x*):= Dx0~bt(x*, to). Then we have the

variational equation

~t(Xo) = Dx f (~ t (Xo , to), to)~t(Xo), ~t0(x0) = I (30)

The Lyapunov exponents are defined by

hi = lim -1 lnlmi(t)l, i = 1 , . . . n (31) t-*oo t

where {mi(t), i = 1, . . . n} are the eigenvalues of,~t(x0). The signs of the Lyapunov exponents provide a quali-

tative assessment of the system dynamics by specifying the long-time average behaviour of the system along the trajectory starting from x0. A positive (negative) Lyapunov exponent along a trajectory in some direction corre- sponds to an expansion (contraction) in that direction along this trajectory. Therefore, intuitively we can see that if the Lyapunov exponents are all strictly negative for a trajectory, then there are contractions or convergence in all directions. That is, this trajectory must be converging to a stable equilibrium point asymptotically. More gen- erally, for any attractor, it is clear that contractions must outweigh expansions in the long-run, and indeed for an attractor, we have

n

~--~ ~,-<0 i=0

Specifically for a stable equilibrium point, as there are only contractions in all directions, all the Lyapunov exponents must be negative ()~i < 0 for all i). For a stable limit cycle, there is invariance along one direction whereas contractions exist in others so that one exponent is zero (~l = 0) and the rest are negative )~i<0 for i = 2 , . . . ,n.

9 For a stable 2-torus which is a two-dimensional limit set, two exponents are zero )~1--)~2 = 0 and the rest are negative )~i < 0 for i---- 3 , . . . , n. Any 'regular' attractor then contains strictly non-positive Lyapunov exponents. Irregular behaviour or chaotic limit behaviour arises when an attractor possesses at least one positive Lyapunov exponent. The positive exponents characterize the fact that there exists stretching or expansions within the

294 Hard-I/mR induced chaos in a fundamental power system model." W. J i and V. Venkatasubramanian

Table 1. Computation of Lyapunov characteristic exponents

Parameters Lyapunov characteristic exponents

PT D A 1 A 2 A 3 A 4

Lyapunov dimensions

dL*

Steady state

0.5 2 -0 .0980 -0 .1095 -0 .8647 1 2 ~ 0 -0.1023 -0 .9195 1.2 2 ~ 0 -0.3547 -0 .3745 1.323 2 0.2945 ~ 0 -0 .3072 1.3232 2 0.2994 ~ 0 -0 .3584 1.32325 2 0.3179 ~ 0 -0.4079

-- 0.8995 0 Equilibrium - 0.9528 1 Periodic - 1.2499 1 Periodic - 1.9976 2.9587 Chaotic - 1.9807 2.8345 Chaotic - 1.9625 2.7794 Chaotic

*From Reference 29.

attractor personifying the chaotic nature of the strange attractors.

For the numerical computation of Lyapunov exponents, we use the method presented in Reference 30 which computes the exponents directly from the differential equation. Since the method in Reference 30 assumes the system to be smooth, we approximate the hard-limit with a smooth function

2 Efd - - 2.5 +-arctan[0.4(Efd r - 2.5)

71"

x exp(0.4(Efd r -- 2.5)2)] (32)

and our choice is purely a matter of convenience. Note that such an approximation amounts to a small perturba- tion of the dynamics E in the space of vector fields. As chaotic motions are persistent under small perturbations in the function space 9, we can expect to see the same phenomena in the system with the smooth approximation as for the actual system E.

The computation results for the Lyapunov exponents are summarized in Table 1 for the power PT variation keeping the damping D = 2 as in Section V.4. The results from Table 1 can be easily compared with the phase- portraits of the various periodic orbits shown in Section V.4 for an analytical verification of the claims in Section V.4 on the stability status of the periodic orbits shown in Figure 20.

We conclude from this table that our system ~ does have chaotic attractors for the Px values PT = 1.323, PT = 1.3232 and Px = 1.32325 as there exists a positive Lyapunov exponent for the respective attractors for each of these parameter values.

As a proof on the existence of chaos for the discussion in Section V.3 under D variation, we compute the Lyapunov exponents for the parameters PT = 1.3 and D---1.485 and the exponents are A1 = 0.25, A 2 ~ 0, A3 = -0.21, and A4 = -1.97. The attractor possesses a positive Lyapunov exponent (A1 = 0.25) which proves that the motion is chaotic in this case.

VI. Conclusion The dynamics of a fundamental power system model in a single-machine infinite-bus representation is studied extensively using numerical simulations and bifurcation theoretic analysis. It is proved that the wind-up l/miters present in the exciter control could induce sustained oscillations after sub-critical Hopf bifurcations. More- over, the hard-limit induced stable limit cycles undergo a sequence of period-doubling bifurcations resulting in chaotic motions. Chaos is shown to exist in this repre-

sentative power system model over a wide range of practical parameter values.

Results in the paper suggest the possible occurrence of chaotic behaviour in the power system from the inter- actions of excitation control dynamics and the hard- limits. A more detailed analysis of the hard-limit induced nonlinear dynamic phenomena in power system models is recommended. The possibility of detecting chaos in more detailed power system models promises to be an interesting and challenging task.

VII. Acknowledgements This research was partially supported by NSF grants ECS-9457126 and ECS-9320041. Support from Bonne- ville Power Administration, Portland, OR under the Power Professorship Program and the support from Union Electric Company, St. Louis, MO under the EPRI research project 3573-10 are also gratefully acknowledged.

VIII. References 1 Anderson, P M and Fouad, A A Power system control and

stability The Iowa State Press, Ames, Iowa (1977)

2 Fink, L H (ed) Proceedings of the NSF International Workshop on Bulk Power System Voltage Phenomena--II, Maryland. CCC Inc., Fairfax, VA (1991)

3 Fink, L H (ed) Proceedings of the NSFInternational Work- shop on Bulk Power System Voltage Phenomena--III, Davos, Switzerland. ECC Inc., Fairfax, VA (1994)

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